ricow@chromium.org | 55ee807 | 2011-09-08 16:33:10 +0000 | [diff] [blame] | 1 | // Copyright 2011 the V8 project authors. All rights reserved. |
erik.corry@gmail.com | 4a6c327 | 2010-11-18 12:04:40 +0000 | [diff] [blame] | 2 | // Redistribution and use in source and binary forms, with or without |
| 3 | // modification, are permitted provided that the following conditions are |
| 4 | // met: |
| 5 | // |
| 6 | // * Redistributions of source code must retain the above copyright |
| 7 | // notice, this list of conditions and the following disclaimer. |
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| 14 | // from this software without specific prior written permission. |
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ulan@chromium.org | 77ca49a | 2013-04-22 09:43:56 +0000 | [diff] [blame] | 28 | #include <cmath> |
erik.corry@gmail.com | 4a6c327 | 2010-11-18 12:04:40 +0000 | [diff] [blame] | 29 | |
ricow@chromium.org | 55ee807 | 2011-09-08 16:33:10 +0000 | [diff] [blame] | 30 | #include "../include/v8stdint.h" |
| 31 | #include "checks.h" |
| 32 | #include "utils.h" |
| 33 | |
erik.corry@gmail.com | 4a6c327 | 2010-11-18 12:04:40 +0000 | [diff] [blame] | 34 | #include "bignum-dtoa.h" |
| 35 | |
| 36 | #include "bignum.h" |
| 37 | #include "double.h" |
| 38 | |
| 39 | namespace v8 { |
| 40 | namespace internal { |
| 41 | |
| 42 | static int NormalizedExponent(uint64_t significand, int exponent) { |
| 43 | ASSERT(significand != 0); |
| 44 | while ((significand & Double::kHiddenBit) == 0) { |
| 45 | significand = significand << 1; |
| 46 | exponent = exponent - 1; |
| 47 | } |
| 48 | return exponent; |
| 49 | } |
| 50 | |
| 51 | |
| 52 | // Forward declarations: |
| 53 | // Returns an estimation of k such that 10^(k-1) <= v < 10^k. |
| 54 | static int EstimatePower(int exponent); |
| 55 | // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator |
| 56 | // and denominator. |
| 57 | static void InitialScaledStartValues(double v, |
| 58 | int estimated_power, |
| 59 | bool need_boundary_deltas, |
| 60 | Bignum* numerator, |
| 61 | Bignum* denominator, |
| 62 | Bignum* delta_minus, |
| 63 | Bignum* delta_plus); |
| 64 | // Multiplies numerator/denominator so that its values lies in the range 1-10. |
| 65 | // Returns decimal_point s.t. |
| 66 | // v = numerator'/denominator' * 10^(decimal_point-1) |
| 67 | // where numerator' and denominator' are the values of numerator and |
| 68 | // denominator after the call to this function. |
| 69 | static void FixupMultiply10(int estimated_power, bool is_even, |
| 70 | int* decimal_point, |
| 71 | Bignum* numerator, Bignum* denominator, |
| 72 | Bignum* delta_minus, Bignum* delta_plus); |
| 73 | // Generates digits from the left to the right and stops when the generated |
| 74 | // digits yield the shortest decimal representation of v. |
| 75 | static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, |
| 76 | Bignum* delta_minus, Bignum* delta_plus, |
| 77 | bool is_even, |
| 78 | Vector<char> buffer, int* length); |
| 79 | // Generates 'requested_digits' after the decimal point. |
| 80 | static void BignumToFixed(int requested_digits, int* decimal_point, |
| 81 | Bignum* numerator, Bignum* denominator, |
| 82 | Vector<char>(buffer), int* length); |
| 83 | // Generates 'count' digits of numerator/denominator. |
| 84 | // Once 'count' digits have been produced rounds the result depending on the |
| 85 | // remainder (remainders of exactly .5 round upwards). Might update the |
| 86 | // decimal_point when rounding up (for example for 0.9999). |
| 87 | static void GenerateCountedDigits(int count, int* decimal_point, |
| 88 | Bignum* numerator, Bignum* denominator, |
| 89 | Vector<char>(buffer), int* length); |
| 90 | |
| 91 | |
| 92 | void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits, |
| 93 | Vector<char> buffer, int* length, int* decimal_point) { |
| 94 | ASSERT(v > 0); |
| 95 | ASSERT(!Double(v).IsSpecial()); |
| 96 | uint64_t significand = Double(v).Significand(); |
| 97 | bool is_even = (significand & 1) == 0; |
| 98 | int exponent = Double(v).Exponent(); |
| 99 | int normalized_exponent = NormalizedExponent(significand, exponent); |
| 100 | // estimated_power might be too low by 1. |
| 101 | int estimated_power = EstimatePower(normalized_exponent); |
| 102 | |
| 103 | // Shortcut for Fixed. |
| 104 | // The requested digits correspond to the digits after the point. If the |
| 105 | // number is much too small, then there is no need in trying to get any |
| 106 | // digits. |
| 107 | if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) { |
| 108 | buffer[0] = '\0'; |
| 109 | *length = 0; |
| 110 | // Set decimal-point to -requested_digits. This is what Gay does. |
| 111 | // Note that it should not have any effect anyways since the string is |
| 112 | // empty. |
| 113 | *decimal_point = -requested_digits; |
| 114 | return; |
| 115 | } |
| 116 | |
| 117 | Bignum numerator; |
| 118 | Bignum denominator; |
| 119 | Bignum delta_minus; |
| 120 | Bignum delta_plus; |
| 121 | // Make sure the bignum can grow large enough. The smallest double equals |
| 122 | // 4e-324. In this case the denominator needs fewer than 324*4 binary digits. |
| 123 | // The maximum double is 1.7976931348623157e308 which needs fewer than |
| 124 | // 308*4 binary digits. |
| 125 | ASSERT(Bignum::kMaxSignificantBits >= 324*4); |
| 126 | bool need_boundary_deltas = (mode == BIGNUM_DTOA_SHORTEST); |
| 127 | InitialScaledStartValues(v, estimated_power, need_boundary_deltas, |
| 128 | &numerator, &denominator, |
| 129 | &delta_minus, &delta_plus); |
| 130 | // We now have v = (numerator / denominator) * 10^estimated_power. |
| 131 | FixupMultiply10(estimated_power, is_even, decimal_point, |
| 132 | &numerator, &denominator, |
| 133 | &delta_minus, &delta_plus); |
| 134 | // We now have v = (numerator / denominator) * 10^(decimal_point-1), and |
| 135 | // 1 <= (numerator + delta_plus) / denominator < 10 |
| 136 | switch (mode) { |
| 137 | case BIGNUM_DTOA_SHORTEST: |
| 138 | GenerateShortestDigits(&numerator, &denominator, |
| 139 | &delta_minus, &delta_plus, |
| 140 | is_even, buffer, length); |
| 141 | break; |
| 142 | case BIGNUM_DTOA_FIXED: |
| 143 | BignumToFixed(requested_digits, decimal_point, |
| 144 | &numerator, &denominator, |
| 145 | buffer, length); |
| 146 | break; |
| 147 | case BIGNUM_DTOA_PRECISION: |
| 148 | GenerateCountedDigits(requested_digits, decimal_point, |
| 149 | &numerator, &denominator, |
| 150 | buffer, length); |
| 151 | break; |
| 152 | default: |
| 153 | UNREACHABLE(); |
| 154 | } |
| 155 | buffer[*length] = '\0'; |
| 156 | } |
| 157 | |
| 158 | |
| 159 | // The procedure starts generating digits from the left to the right and stops |
| 160 | // when the generated digits yield the shortest decimal representation of v. A |
| 161 | // decimal representation of v is a number lying closer to v than to any other |
| 162 | // double, so it converts to v when read. |
| 163 | // |
| 164 | // This is true if d, the decimal representation, is between m- and m+, the |
| 165 | // upper and lower boundaries. d must be strictly between them if !is_even. |
| 166 | // m- := (numerator - delta_minus) / denominator |
| 167 | // m+ := (numerator + delta_plus) / denominator |
| 168 | // |
| 169 | // Precondition: 0 <= (numerator+delta_plus) / denominator < 10. |
| 170 | // If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit |
| 171 | // will be produced. This should be the standard precondition. |
| 172 | static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, |
| 173 | Bignum* delta_minus, Bignum* delta_plus, |
| 174 | bool is_even, |
| 175 | Vector<char> buffer, int* length) { |
| 176 | // Small optimization: if delta_minus and delta_plus are the same just reuse |
| 177 | // one of the two bignums. |
| 178 | if (Bignum::Equal(*delta_minus, *delta_plus)) { |
| 179 | delta_plus = delta_minus; |
| 180 | } |
| 181 | *length = 0; |
| 182 | while (true) { |
| 183 | uint16_t digit; |
| 184 | digit = numerator->DivideModuloIntBignum(*denominator); |
| 185 | ASSERT(digit <= 9); // digit is a uint16_t and therefore always positive. |
| 186 | // digit = numerator / denominator (integer division). |
| 187 | // numerator = numerator % denominator. |
| 188 | buffer[(*length)++] = digit + '0'; |
| 189 | |
| 190 | // Can we stop already? |
| 191 | // If the remainder of the division is less than the distance to the lower |
| 192 | // boundary we can stop. In this case we simply round down (discarding the |
| 193 | // remainder). |
| 194 | // Similarly we test if we can round up (using the upper boundary). |
| 195 | bool in_delta_room_minus; |
| 196 | bool in_delta_room_plus; |
| 197 | if (is_even) { |
| 198 | in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus); |
| 199 | } else { |
| 200 | in_delta_room_minus = Bignum::Less(*numerator, *delta_minus); |
| 201 | } |
| 202 | if (is_even) { |
| 203 | in_delta_room_plus = |
| 204 | Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; |
| 205 | } else { |
| 206 | in_delta_room_plus = |
| 207 | Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; |
| 208 | } |
| 209 | if (!in_delta_room_minus && !in_delta_room_plus) { |
| 210 | // Prepare for next iteration. |
| 211 | numerator->Times10(); |
| 212 | delta_minus->Times10(); |
| 213 | // We optimized delta_plus to be equal to delta_minus (if they share the |
| 214 | // same value). So don't multiply delta_plus if they point to the same |
| 215 | // object. |
| 216 | if (delta_minus != delta_plus) { |
| 217 | delta_plus->Times10(); |
| 218 | } |
| 219 | } else if (in_delta_room_minus && in_delta_room_plus) { |
| 220 | // Let's see if 2*numerator < denominator. |
| 221 | // If yes, then the next digit would be < 5 and we can round down. |
| 222 | int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator); |
| 223 | if (compare < 0) { |
| 224 | // Remaining digits are less than .5. -> Round down (== do nothing). |
| 225 | } else if (compare > 0) { |
| 226 | // Remaining digits are more than .5 of denominator. -> Round up. |
| 227 | // Note that the last digit could not be a '9' as otherwise the whole |
| 228 | // loop would have stopped earlier. |
| 229 | // We still have an assert here in case the preconditions were not |
| 230 | // satisfied. |
| 231 | ASSERT(buffer[(*length) - 1] != '9'); |
| 232 | buffer[(*length) - 1]++; |
| 233 | } else { |
| 234 | // Halfway case. |
| 235 | // TODO(floitsch): need a way to solve half-way cases. |
| 236 | // For now let's round towards even (since this is what Gay seems to |
| 237 | // do). |
| 238 | |
| 239 | if ((buffer[(*length) - 1] - '0') % 2 == 0) { |
| 240 | // Round down => Do nothing. |
| 241 | } else { |
| 242 | ASSERT(buffer[(*length) - 1] != '9'); |
| 243 | buffer[(*length) - 1]++; |
| 244 | } |
| 245 | } |
| 246 | return; |
| 247 | } else if (in_delta_room_minus) { |
| 248 | // Round down (== do nothing). |
| 249 | return; |
| 250 | } else { // in_delta_room_plus |
| 251 | // Round up. |
| 252 | // Note again that the last digit could not be '9' since this would have |
| 253 | // stopped the loop earlier. |
| 254 | // We still have an ASSERT here, in case the preconditions were not |
| 255 | // satisfied. |
| 256 | ASSERT(buffer[(*length) -1] != '9'); |
| 257 | buffer[(*length) - 1]++; |
| 258 | return; |
| 259 | } |
| 260 | } |
| 261 | } |
| 262 | |
| 263 | |
| 264 | // Let v = numerator / denominator < 10. |
| 265 | // Then we generate 'count' digits of d = x.xxxxx... (without the decimal point) |
| 266 | // from left to right. Once 'count' digits have been produced we decide wether |
| 267 | // to round up or down. Remainders of exactly .5 round upwards. Numbers such |
| 268 | // as 9.999999 propagate a carry all the way, and change the |
| 269 | // exponent (decimal_point), when rounding upwards. |
| 270 | static void GenerateCountedDigits(int count, int* decimal_point, |
| 271 | Bignum* numerator, Bignum* denominator, |
| 272 | Vector<char>(buffer), int* length) { |
| 273 | ASSERT(count >= 0); |
| 274 | for (int i = 0; i < count - 1; ++i) { |
| 275 | uint16_t digit; |
| 276 | digit = numerator->DivideModuloIntBignum(*denominator); |
| 277 | ASSERT(digit <= 9); // digit is a uint16_t and therefore always positive. |
| 278 | // digit = numerator / denominator (integer division). |
| 279 | // numerator = numerator % denominator. |
| 280 | buffer[i] = digit + '0'; |
| 281 | // Prepare for next iteration. |
| 282 | numerator->Times10(); |
| 283 | } |
| 284 | // Generate the last digit. |
| 285 | uint16_t digit; |
| 286 | digit = numerator->DivideModuloIntBignum(*denominator); |
| 287 | if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { |
| 288 | digit++; |
| 289 | } |
| 290 | buffer[count - 1] = digit + '0'; |
| 291 | // Correct bad digits (in case we had a sequence of '9's). Propagate the |
| 292 | // carry until we hat a non-'9' or til we reach the first digit. |
| 293 | for (int i = count - 1; i > 0; --i) { |
| 294 | if (buffer[i] != '0' + 10) break; |
| 295 | buffer[i] = '0'; |
| 296 | buffer[i - 1]++; |
| 297 | } |
| 298 | if (buffer[0] == '0' + 10) { |
| 299 | // Propagate a carry past the top place. |
| 300 | buffer[0] = '1'; |
| 301 | (*decimal_point)++; |
| 302 | } |
| 303 | *length = count; |
| 304 | } |
| 305 | |
| 306 | |
| 307 | // Generates 'requested_digits' after the decimal point. It might omit |
| 308 | // trailing '0's. If the input number is too small then no digits at all are |
| 309 | // generated (ex.: 2 fixed digits for 0.00001). |
| 310 | // |
| 311 | // Input verifies: 1 <= (numerator + delta) / denominator < 10. |
| 312 | static void BignumToFixed(int requested_digits, int* decimal_point, |
| 313 | Bignum* numerator, Bignum* denominator, |
| 314 | Vector<char>(buffer), int* length) { |
| 315 | // Note that we have to look at more than just the requested_digits, since |
| 316 | // a number could be rounded up. Example: v=0.5 with requested_digits=0. |
| 317 | // Even though the power of v equals 0 we can't just stop here. |
| 318 | if (-(*decimal_point) > requested_digits) { |
| 319 | // The number is definitively too small. |
| 320 | // Ex: 0.001 with requested_digits == 1. |
| 321 | // Set decimal-point to -requested_digits. This is what Gay does. |
| 322 | // Note that it should not have any effect anyways since the string is |
| 323 | // empty. |
| 324 | *decimal_point = -requested_digits; |
| 325 | *length = 0; |
| 326 | return; |
| 327 | } else if (-(*decimal_point) == requested_digits) { |
| 328 | // We only need to verify if the number rounds down or up. |
| 329 | // Ex: 0.04 and 0.06 with requested_digits == 1. |
| 330 | ASSERT(*decimal_point == -requested_digits); |
| 331 | // Initially the fraction lies in range (1, 10]. Multiply the denominator |
| 332 | // by 10 so that we can compare more easily. |
| 333 | denominator->Times10(); |
| 334 | if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { |
| 335 | // If the fraction is >= 0.5 then we have to include the rounded |
| 336 | // digit. |
| 337 | buffer[0] = '1'; |
| 338 | *length = 1; |
| 339 | (*decimal_point)++; |
| 340 | } else { |
| 341 | // Note that we caught most of similar cases earlier. |
| 342 | *length = 0; |
| 343 | } |
| 344 | return; |
| 345 | } else { |
| 346 | // The requested digits correspond to the digits after the point. |
| 347 | // The variable 'needed_digits' includes the digits before the point. |
| 348 | int needed_digits = (*decimal_point) + requested_digits; |
| 349 | GenerateCountedDigits(needed_digits, decimal_point, |
| 350 | numerator, denominator, |
| 351 | buffer, length); |
| 352 | } |
| 353 | } |
| 354 | |
| 355 | |
| 356 | // Returns an estimation of k such that 10^(k-1) <= v < 10^k where |
| 357 | // v = f * 2^exponent and 2^52 <= f < 2^53. |
| 358 | // v is hence a normalized double with the given exponent. The output is an |
| 359 | // approximation for the exponent of the decimal approimation .digits * 10^k. |
| 360 | // |
| 361 | // The result might undershoot by 1 in which case 10^k <= v < 10^k+1. |
| 362 | // Note: this property holds for v's upper boundary m+ too. |
| 363 | // 10^k <= m+ < 10^k+1. |
| 364 | // (see explanation below). |
| 365 | // |
| 366 | // Examples: |
| 367 | // EstimatePower(0) => 16 |
| 368 | // EstimatePower(-52) => 0 |
| 369 | // |
| 370 | // Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0. |
| 371 | static int EstimatePower(int exponent) { |
| 372 | // This function estimates log10 of v where v = f*2^e (with e == exponent). |
| 373 | // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)). |
| 374 | // Note that f is bounded by its container size. Let p = 53 (the double's |
| 375 | // significand size). Then 2^(p-1) <= f < 2^p. |
| 376 | // |
| 377 | // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close |
| 378 | // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)). |
| 379 | // The computed number undershoots by less than 0.631 (when we compute log3 |
| 380 | // and not log10). |
| 381 | // |
| 382 | // Optimization: since we only need an approximated result this computation |
| 383 | // can be performed on 64 bit integers. On x86/x64 architecture the speedup is |
| 384 | // not really measurable, though. |
| 385 | // |
| 386 | // Since we want to avoid overshooting we decrement by 1e10 so that |
| 387 | // floating-point imprecisions don't affect us. |
| 388 | // |
| 389 | // Explanation for v's boundary m+: the computation takes advantage of |
| 390 | // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement |
| 391 | // (even for denormals where the delta can be much more important). |
| 392 | |
| 393 | const double k1Log10 = 0.30102999566398114; // 1/lg(10) |
| 394 | |
| 395 | // For doubles len(f) == 53 (don't forget the hidden bit). |
| 396 | const int kSignificandSize = 53; |
| 397 | double estimate = ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10); |
| 398 | return static_cast<int>(estimate); |
| 399 | } |
| 400 | |
| 401 | |
| 402 | // See comments for InitialScaledStartValues. |
| 403 | static void InitialScaledStartValuesPositiveExponent( |
| 404 | double v, int estimated_power, bool need_boundary_deltas, |
| 405 | Bignum* numerator, Bignum* denominator, |
| 406 | Bignum* delta_minus, Bignum* delta_plus) { |
| 407 | // A positive exponent implies a positive power. |
| 408 | ASSERT(estimated_power >= 0); |
| 409 | // Since the estimated_power is positive we simply multiply the denominator |
| 410 | // by 10^estimated_power. |
| 411 | |
| 412 | // numerator = v. |
| 413 | numerator->AssignUInt64(Double(v).Significand()); |
| 414 | numerator->ShiftLeft(Double(v).Exponent()); |
| 415 | // denominator = 10^estimated_power. |
| 416 | denominator->AssignPowerUInt16(10, estimated_power); |
| 417 | |
| 418 | if (need_boundary_deltas) { |
| 419 | // Introduce a common denominator so that the deltas to the boundaries are |
| 420 | // integers. |
| 421 | denominator->ShiftLeft(1); |
| 422 | numerator->ShiftLeft(1); |
| 423 | // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common |
| 424 | // denominator (of 2) delta_plus equals 2^e. |
| 425 | delta_plus->AssignUInt16(1); |
| 426 | delta_plus->ShiftLeft(Double(v).Exponent()); |
| 427 | // Same for delta_minus (with adjustments below if f == 2^p-1). |
| 428 | delta_minus->AssignUInt16(1); |
| 429 | delta_minus->ShiftLeft(Double(v).Exponent()); |
| 430 | |
| 431 | // If the significand (without the hidden bit) is 0, then the lower |
| 432 | // boundary is closer than just half a ulp (unit in the last place). |
| 433 | // There is only one exception: if the next lower number is a denormal then |
| 434 | // the distance is 1 ulp. This cannot be the case for exponent >= 0 (but we |
| 435 | // have to test it in the other function where exponent < 0). |
| 436 | uint64_t v_bits = Double(v).AsUint64(); |
| 437 | if ((v_bits & Double::kSignificandMask) == 0) { |
| 438 | // The lower boundary is closer at half the distance of "normal" numbers. |
| 439 | // Increase the common denominator and adapt all but the delta_minus. |
| 440 | denominator->ShiftLeft(1); // *2 |
| 441 | numerator->ShiftLeft(1); // *2 |
| 442 | delta_plus->ShiftLeft(1); // *2 |
| 443 | } |
| 444 | } |
| 445 | } |
| 446 | |
| 447 | |
| 448 | // See comments for InitialScaledStartValues |
| 449 | static void InitialScaledStartValuesNegativeExponentPositivePower( |
| 450 | double v, int estimated_power, bool need_boundary_deltas, |
| 451 | Bignum* numerator, Bignum* denominator, |
| 452 | Bignum* delta_minus, Bignum* delta_plus) { |
| 453 | uint64_t significand = Double(v).Significand(); |
| 454 | int exponent = Double(v).Exponent(); |
| 455 | // v = f * 2^e with e < 0, and with estimated_power >= 0. |
| 456 | // This means that e is close to 0 (have a look at how estimated_power is |
| 457 | // computed). |
| 458 | |
| 459 | // numerator = significand |
| 460 | // since v = significand * 2^exponent this is equivalent to |
| 461 | // numerator = v * / 2^-exponent |
| 462 | numerator->AssignUInt64(significand); |
| 463 | // denominator = 10^estimated_power * 2^-exponent (with exponent < 0) |
| 464 | denominator->AssignPowerUInt16(10, estimated_power); |
| 465 | denominator->ShiftLeft(-exponent); |
| 466 | |
| 467 | if (need_boundary_deltas) { |
| 468 | // Introduce a common denominator so that the deltas to the boundaries are |
| 469 | // integers. |
| 470 | denominator->ShiftLeft(1); |
| 471 | numerator->ShiftLeft(1); |
| 472 | // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common |
| 473 | // denominator (of 2) delta_plus equals 2^e. |
| 474 | // Given that the denominator already includes v's exponent the distance |
| 475 | // to the boundaries is simply 1. |
| 476 | delta_plus->AssignUInt16(1); |
| 477 | // Same for delta_minus (with adjustments below if f == 2^p-1). |
| 478 | delta_minus->AssignUInt16(1); |
| 479 | |
| 480 | // If the significand (without the hidden bit) is 0, then the lower |
| 481 | // boundary is closer than just one ulp (unit in the last place). |
| 482 | // There is only one exception: if the next lower number is a denormal |
| 483 | // then the distance is 1 ulp. Since the exponent is close to zero |
| 484 | // (otherwise estimated_power would have been negative) this cannot happen |
| 485 | // here either. |
| 486 | uint64_t v_bits = Double(v).AsUint64(); |
| 487 | if ((v_bits & Double::kSignificandMask) == 0) { |
| 488 | // The lower boundary is closer at half the distance of "normal" numbers. |
| 489 | // Increase the denominator and adapt all but the delta_minus. |
| 490 | denominator->ShiftLeft(1); // *2 |
| 491 | numerator->ShiftLeft(1); // *2 |
| 492 | delta_plus->ShiftLeft(1); // *2 |
| 493 | } |
| 494 | } |
| 495 | } |
| 496 | |
| 497 | |
| 498 | // See comments for InitialScaledStartValues |
| 499 | static void InitialScaledStartValuesNegativeExponentNegativePower( |
| 500 | double v, int estimated_power, bool need_boundary_deltas, |
| 501 | Bignum* numerator, Bignum* denominator, |
| 502 | Bignum* delta_minus, Bignum* delta_plus) { |
| 503 | const uint64_t kMinimalNormalizedExponent = |
| 504 | V8_2PART_UINT64_C(0x00100000, 00000000); |
| 505 | uint64_t significand = Double(v).Significand(); |
| 506 | int exponent = Double(v).Exponent(); |
| 507 | // Instead of multiplying the denominator with 10^estimated_power we |
| 508 | // multiply all values (numerator and deltas) by 10^-estimated_power. |
| 509 | |
| 510 | // Use numerator as temporary container for power_ten. |
| 511 | Bignum* power_ten = numerator; |
| 512 | power_ten->AssignPowerUInt16(10, -estimated_power); |
| 513 | |
| 514 | if (need_boundary_deltas) { |
| 515 | // Since power_ten == numerator we must make a copy of 10^estimated_power |
| 516 | // before we complete the computation of the numerator. |
| 517 | // delta_plus = delta_minus = 10^estimated_power |
| 518 | delta_plus->AssignBignum(*power_ten); |
| 519 | delta_minus->AssignBignum(*power_ten); |
| 520 | } |
| 521 | |
| 522 | // numerator = significand * 2 * 10^-estimated_power |
| 523 | // since v = significand * 2^exponent this is equivalent to |
| 524 | // numerator = v * 10^-estimated_power * 2 * 2^-exponent. |
| 525 | // Remember: numerator has been abused as power_ten. So no need to assign it |
| 526 | // to itself. |
| 527 | ASSERT(numerator == power_ten); |
| 528 | numerator->MultiplyByUInt64(significand); |
| 529 | |
| 530 | // denominator = 2 * 2^-exponent with exponent < 0. |
| 531 | denominator->AssignUInt16(1); |
| 532 | denominator->ShiftLeft(-exponent); |
| 533 | |
| 534 | if (need_boundary_deltas) { |
| 535 | // Introduce a common denominator so that the deltas to the boundaries are |
| 536 | // integers. |
| 537 | numerator->ShiftLeft(1); |
| 538 | denominator->ShiftLeft(1); |
| 539 | // With this shift the boundaries have their correct value, since |
| 540 | // delta_plus = 10^-estimated_power, and |
| 541 | // delta_minus = 10^-estimated_power. |
| 542 | // These assignments have been done earlier. |
| 543 | |
| 544 | // The special case where the lower boundary is twice as close. |
| 545 | // This time we have to look out for the exception too. |
| 546 | uint64_t v_bits = Double(v).AsUint64(); |
| 547 | if ((v_bits & Double::kSignificandMask) == 0 && |
| 548 | // The only exception where a significand == 0 has its boundaries at |
| 549 | // "normal" distances: |
| 550 | (v_bits & Double::kExponentMask) != kMinimalNormalizedExponent) { |
| 551 | numerator->ShiftLeft(1); // *2 |
| 552 | denominator->ShiftLeft(1); // *2 |
| 553 | delta_plus->ShiftLeft(1); // *2 |
| 554 | } |
| 555 | } |
| 556 | } |
| 557 | |
| 558 | |
| 559 | // Let v = significand * 2^exponent. |
| 560 | // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator |
| 561 | // and denominator. The functions GenerateShortestDigits and |
| 562 | // GenerateCountedDigits will then convert this ratio to its decimal |
| 563 | // representation d, with the required accuracy. |
| 564 | // Then d * 10^estimated_power is the representation of v. |
| 565 | // (Note: the fraction and the estimated_power might get adjusted before |
| 566 | // generating the decimal representation.) |
| 567 | // |
| 568 | // The initial start values consist of: |
| 569 | // - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power. |
| 570 | // - a scaled (common) denominator. |
| 571 | // optionally (used by GenerateShortestDigits to decide if it has the shortest |
| 572 | // decimal converting back to v): |
| 573 | // - v - m-: the distance to the lower boundary. |
| 574 | // - m+ - v: the distance to the upper boundary. |
| 575 | // |
| 576 | // v, m+, m-, and therefore v - m- and m+ - v all share the same denominator. |
| 577 | // |
| 578 | // Let ep == estimated_power, then the returned values will satisfy: |
| 579 | // v / 10^ep = numerator / denominator. |
| 580 | // v's boundarys m- and m+: |
| 581 | // m- / 10^ep == v / 10^ep - delta_minus / denominator |
| 582 | // m+ / 10^ep == v / 10^ep + delta_plus / denominator |
| 583 | // Or in other words: |
| 584 | // m- == v - delta_minus * 10^ep / denominator; |
| 585 | // m+ == v + delta_plus * 10^ep / denominator; |
| 586 | // |
| 587 | // Since 10^(k-1) <= v < 10^k (with k == estimated_power) |
| 588 | // or 10^k <= v < 10^(k+1) |
| 589 | // we then have 0.1 <= numerator/denominator < 1 |
| 590 | // or 1 <= numerator/denominator < 10 |
| 591 | // |
| 592 | // It is then easy to kickstart the digit-generation routine. |
| 593 | // |
| 594 | // The boundary-deltas are only filled if need_boundary_deltas is set. |
| 595 | static void InitialScaledStartValues(double v, |
| 596 | int estimated_power, |
| 597 | bool need_boundary_deltas, |
| 598 | Bignum* numerator, |
| 599 | Bignum* denominator, |
| 600 | Bignum* delta_minus, |
| 601 | Bignum* delta_plus) { |
| 602 | if (Double(v).Exponent() >= 0) { |
| 603 | InitialScaledStartValuesPositiveExponent( |
| 604 | v, estimated_power, need_boundary_deltas, |
| 605 | numerator, denominator, delta_minus, delta_plus); |
| 606 | } else if (estimated_power >= 0) { |
| 607 | InitialScaledStartValuesNegativeExponentPositivePower( |
| 608 | v, estimated_power, need_boundary_deltas, |
| 609 | numerator, denominator, delta_minus, delta_plus); |
| 610 | } else { |
| 611 | InitialScaledStartValuesNegativeExponentNegativePower( |
| 612 | v, estimated_power, need_boundary_deltas, |
| 613 | numerator, denominator, delta_minus, delta_plus); |
| 614 | } |
| 615 | } |
| 616 | |
| 617 | |
| 618 | // This routine multiplies numerator/denominator so that its values lies in the |
| 619 | // range 1-10. That is after a call to this function we have: |
| 620 | // 1 <= (numerator + delta_plus) /denominator < 10. |
| 621 | // Let numerator the input before modification and numerator' the argument |
| 622 | // after modification, then the output-parameter decimal_point is such that |
| 623 | // numerator / denominator * 10^estimated_power == |
| 624 | // numerator' / denominator' * 10^(decimal_point - 1) |
| 625 | // In some cases estimated_power was too low, and this is already the case. We |
| 626 | // then simply adjust the power so that 10^(k-1) <= v < 10^k (with k == |
| 627 | // estimated_power) but do not touch the numerator or denominator. |
| 628 | // Otherwise the routine multiplies the numerator and the deltas by 10. |
| 629 | static void FixupMultiply10(int estimated_power, bool is_even, |
| 630 | int* decimal_point, |
| 631 | Bignum* numerator, Bignum* denominator, |
| 632 | Bignum* delta_minus, Bignum* delta_plus) { |
| 633 | bool in_range; |
| 634 | if (is_even) { |
| 635 | // For IEEE doubles half-way cases (in decimal system numbers ending with 5) |
| 636 | // are rounded to the closest floating-point number with even significand. |
| 637 | in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; |
| 638 | } else { |
| 639 | in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; |
| 640 | } |
| 641 | if (in_range) { |
| 642 | // Since numerator + delta_plus >= denominator we already have |
| 643 | // 1 <= numerator/denominator < 10. Simply update the estimated_power. |
| 644 | *decimal_point = estimated_power + 1; |
| 645 | } else { |
| 646 | *decimal_point = estimated_power; |
| 647 | numerator->Times10(); |
| 648 | if (Bignum::Equal(*delta_minus, *delta_plus)) { |
| 649 | delta_minus->Times10(); |
| 650 | delta_plus->AssignBignum(*delta_minus); |
| 651 | } else { |
| 652 | delta_minus->Times10(); |
| 653 | delta_plus->Times10(); |
| 654 | } |
| 655 | } |
| 656 | } |
| 657 | |
| 658 | } } // namespace v8::internal |